Question:
For any two sets, prove that:
(i) $A \cup(A \cap B)=A$
(ii) $A \cap(A \cup B)=A$
Solution:
(i)
$\mathrm{LHS}=A \cup(A \cap B)$
$\Rightarrow \mathrm{LHS}=(A \cup A) \cap(A \cup B)$
$\Rightarrow \mathrm{LHS}=A \cap(A \cup B) \quad(\because A \subset A \cup B)$
$\Rightarrow \mathrm{LHS}=A=\mathrm{RHS}$
(ii)
$\mathrm{LHS}=A \cap(A \cup B)$
$\Rightarrow \mathrm{LHS}=(A \cap A) \cup(A \cap B)$
$\Rightarrow \mathrm{LHS}=A \cup(A \cap B)$
$\Rightarrow \mathrm{LHS}=A=\mathrm{RHS}$
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